SOLVING COMPLEX TRIGONOMETRIC EQUATIONS
Copyright by Ingrid Stewart, Ph.D.  Please Send Questions and Comments to ingrid.stewart@csn.edu.

Learning Objectives - This is what you must know after studying the lecture and doing the practice problems!

1.   Solve equations containing squared trigonometric ratios.
2.   Solve trigonometric equations involving multiples of an angle.

In the previous lesson, we learned how to solve simple trigonometric equations. Now we will look at a few additional types of equations which are more complex.

Solve Equations containing Squared Trigonometric Ratios - see #1 through 5 in the "Examples" document

Solution Strategy:

Step 1:

If necessary, change all trigonometric ratios in the equation to the same ratio. That is, the equation must contain all sines, cosines, tangents, etc. Apply the appropriate identities to accomplish this!

Step 2:

Either factor the trigonometric equation and apply the Zero Product Principle* or use the Square Root Property**. We will end up with one or more simple trigonometric equations.

* The Zero Product Principle states that if A and B are two expressions and multiplication of A and B equals 0, then either A must be equal to 0 or B must be equal to 0 or both are equal to 0.

** The Square Root Property states that if , then . Please note that u can be any algebraic expression containing any variable and d is a constant. The coefficient of u² must be 1.

Steps 3 through 6

Find the solutions for the angle in all simple trigonometric equations found in Step 2 on the given solution interval. Use Steps 1 through 4 from Lesson 12 to solve the simple trigonometric equations.

Solve Trigonometric Equations Involving Multiples of an Angle - see #6 and 7 in the "Examples" document

Solution Strategy for solving equations of the form

asin(bx) = C

acos(bx) = C

atan(bx) = C 

where b is NOT 1! This is unlike Lesson 12 where we worked strictly with equations in which b was equal to 1.

Step 1:

Find the solution interval for angle bx. Since we are always given a solution interval for the angle x, we simply multiply its end points by the number b.

Please note that we will work with the angle bx through Step 5. Only in Step 6 will we find the solution(s) for angle x.

Steps 2 through 5:

Find the solutions for angle bx on its interval calculated in Step 1. Use Steps 1 through 4 from Lesson 12 - Solving Simple Trigonometric Equations.

Step 6:

Find the solutions for angle x. Once we have found the solutions for angle bx, we now divide/multiply them by b.